On graph-restrictive permutation groups

TL;DR

This paper explores the concept of graph-restrictive permutation groups, proving that such groups must be semiprimitive, and proposes a strong generalization of the Weiss Conjecture linking graph-restrictiveness to semiprimitivity.

Contribution

It proves that all graph-restrictive groups are semiprimitive and proposes a conjecture that characterizes graph-restrictive groups as exactly the semiprimitive ones.

Findings

Proved that graph-restrictive groups are semiprimitive.

Collected and proved results related to the converse implication.

Proposed a strong generalization of the Weiss Conjecture.

Abstract

Let $\Gamma$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex of $\Gamma$ and let $L=G_v^{\Gamma(v)}$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. Then $(\Gamma,G)$ is said to be \emph{locally-$L$}. A transitive permutation group $L$ is \emph{graph-restrictive} if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(\Gamma,G)$ and an arc $(u,v)$ of $\Gamma$, the inequality $|G_{uv}|\leq c(L)$ holds. Using this terminology, the Weiss Conjecture says that primitive groups are graph-restrictive. We propose a very strong generalisation of this conjecture: a group is graph-restrictive if and only if it is semiprimitive. (A transitive permutation group is said to be \emph{semiprimitive} if each of its normal subgroups is either transitive or semiregular.) Our main result is a proof of one of the two implications of this conjecture, namely that graph-restrictive groups are semiprimitive. We also collect the known results and prove some new ones regarding the other implication.